The notion of a P-ultrafilter on omega can be straightened in a natural way to the notion of a coherent P-ultrafilter on a complete ccc Boolean algebra. These ultrafilters exist generically under the condition isolated by Ketonen, namely c = d.
Similarly, under the Canjar condition c = cov(Meager), coherently Ramsey ultrafilters can be shown to exist.
Existence of “coherent” versions of other traditional objects is an ongoing programme.
The coherent ultrafilters are relevant in an old topological question: a coherent P-ultrafilter on an algebra B is an untouchable point in the Stone space of B, witnessing its nonhomogeneity.